MinT - Best Known (86, 237, s)-Nets in Base 3

Best Known (86, 237, s)-Nets in Base 3

(86, 237, 61)-Net over F₃ — Constructive and digital

Digital (86, 237, 61)-net over F₃, using

net from sequence [i] based on digital (86, 60)-sequence over F₃, using
- base reduction for sequences [i] based on digital (13, 60)-sequence over F₉, using
  - s-reduction based on digital (13, 63)-sequence over F₉, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 13 and N(F) ≥ 64, using
      - a function field by GarcÃa and Quoos [i]

(86, 237, 84)-Net over F₃ — Digital

Digital (86, 237, 84)-net over F₃, using

t-expansion [i] based on digital (71, 237, 84)-net over F₃, using
- net from sequence [i] based on digital (71, 83)-sequence over F₃, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 71 and N(F) ≥ 84, using
    - algebraic function fields over â„¤₃ by Niederreiter/Xing [i]

(86, 237, 379)-Net over F₃ — Upper bound on s (digital)

There is no digital (86, 237, 380)-net over F₃, because

1 times m-reduction [i] would yield digital (86, 236, 380)-net over F₃, but
- extracting embedded orthogonal array [i] would yield linear OA(3²³⁶, 380, F₃, 150) (dual of [380, 144, 151]-code), but
  - residual code [i] would yield linear OA(3⁸⁶, 229, F₃, 50) (dual of [229, 143, 51]-code), but
    - the Johnson bound shows that NÂ â‰¤ 153 153734 778487 159186 543991 832667 435898 157821 534223 513753 575995 914628 < 3¹⁴³ [i]

(86, 237, 386)-Net in Base 3 — Upper bound on s

There is no (86, 237, 387)-net in base 3, because

1 times m-reduction [i] would yield (86, 236, 387)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 46309 353553 590362 584283 122381 895866 860108 295254 223479 481636 558692 655218 946588 071857 471678 847015 609126 466356 965787 > 3²³⁶ [i]